University:
Whitman College
Course:
MATH-126 | Calculus II
Academic year:

2024
Views:

320

Pages:

2
Author:

bendypennisetum

Review Problems: Chapter 11 1. What does it mean to say that a series “converges” (I’m looking for the definition; be sure you define any notation you use). 2. Does the given sequence or series converge or diverge? ∞ X 1 √ (a) n n=2 n − (b) n n√ 1+ n (e) ∞ X (−6) n−1 1−n 5 (j) n=1 (f) o n n! (n+2)! ∞ X o 1 · 3 · 5 · · · (2n − 1) 5n n! n=1 ∞ X 3n + 2n (h) 6n n=2 (g) ∞ X n2 + 1 (c) 3 n=2 n − 1 √ ∞ X 5−2 n (d) n3 n=1 (i) n sin nπ 2 o (k) (l) (m) ∞ X 1 n=1 n(n + 1)(n + 2) ∞ X sin2 (n) √ n=1 n n ∞ X (−5)2n 2 n n=1 n 9 ∞ X n n n=1 (n + 1)2 3. Find the sum of the series (a) ∞ X (−3)n−1 22n n=1 (b) ∞ X (x − 3)2n 3n n=2 (c) ∞ X (−1)n π 2n 2n n=0 3 (2n!) Hint for 2(c): Does the series look familiar? 4. Find the radius of convergence. For the last two, include the interval of convergence. (a) X n!xn 1 · 3 · 5 · · · (2n − 1) (b) ∞ X (−1)n n=0 xn n2 5n (c) ∞ X 2n (x − 3)n n=0 √ n+3 sin(x) − x x→0 x3 5. Use a series to evaluate the following limit: lim 6. Use a known template series to find a series for the following: (a) x2 1+x (b) 10x (c) xe2x x Hint for 5(b): 10x = eln(10 ) = ex ln(10) 7. Find the Taylor series for f (x) centered at the given base point: (a) x4 − 3x2 + 1, at x = 1 √ (b) 1/ x at x = 9 (just get the first four non-zero terms of the power series). (c) x−2 at x = 1. In this case, find a pattern for the nth coefficient so that you can write the general series. Using this answer, find the radius of convergence. 8. True or False, and give a short reason: (a) If limn→∞ an = 0, then the series P an is convergent. 1 (b) If P cn 6n is convergent, so is P cn (−2)n . (c) The Ratio Test can be used to determine if a p−series is convergent. (d) If 0 ≤ an ≤ bn and P bn diverges, then P an diverges. (e) 0.9999999 . . . = 1 (f) If an > 0 and P an converges, then (−1)n an converges. P (g) If f (x) = 2x − x2 + 31 x3 − · · · converges for all x, then f 000 (0) = 2. n 9. Suppose that ∞ n=0 cn (x − 1) converges when x = 3 and diverges when x = −2. What can be said about the convergence or divergence of the following? P (a) P cn (b) 10. Find the sum: P∞ n=2 n(n−1) 2n (−1)n cn P (c) P cn 3n Hint: Use the geometric series and the derivative 11. Find the Maclaurin series for ln(x + 1) and find the radius of convergence. 12. Find the sum: 3 + 13. Let an = 9 2! + 27 3! + 81 4! + ··· 2n 3n+1 (a) Determine whether {an } is convergent. (b) Determine whether P∞ n=1 an is convergent. 14. Same as the previous problem, but use an = 1 + 2n 3n 15. Compute the sum 1 + a + a2 + a3 + a4 + a5 + a6 + a7 . Hint: Your answer should be in terms of a fraction. You might start with s equal to the expression above. 16. Explain the difference between absolute and conditional convergence. Which is “better” and why? 2

Calculus II Review Problems: Chapter 11 - Assignment

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